9 THE PLANCHEREL-GODEMENT THEOREM 373 such that for all x e A the function X}~^^(x) = %(x) belongs to &&S, m) n <jfg(S). f(x, y) = f x(%)$(x) dm(%) is a bitrace on A satisfying (U) and (N), Jo ' A/it0, tf/7<s? «5/^ are separable; and we have Sg> = S andmg< = m. The proof of this theorem is in several steps. (15.9.2.2) Construction of $g and proof of (ii). The subalgebra C • 1H^ 4- &tg = s/g is closed in J§f(H5) (5.9.2) and hence is a commutative star algebra with unit element. For each character £' e X(jaQ, it follows from (15.4.14) that % o £7^ is identically zero on A or else is a hermitian character of A: in other words, co: £'»-»•£;' ° Ug is a mapping of X(jtfp into H(A) u {0}. This mapping co is infective, because we have ^'Ofy) == * f°r a^ £' e X(cS^), and since £' is continuous on ^, the values of £' on Ug(A), which is dense in j/^, completely determine the character <f. On the other hand, it follows from the definitions that co is continuous with respect to the weak topologies on X(jfg) and CA; since X(s/g) is metrizable and compact, the same is true of its image o>(X(j*;)) = S; c: H(A) u {0}, and co is a homeomorphism of X(jtfg) onto S^ (12.3.6). If 1H €«i/5, then js/^ = cf/0, and S^ does not contain the element 0 of CA. In this case we put S^ = S^. If on the other hand 1H ^ jtfg9 then stfg is a closed hyperplane and an ideal in j^, hence is a maximal ideal, and there exists a character £'Q of jaf^ whose kernel is jtfg (15.3.1); its image under co is therefore the element 0 of CA, and we put S^ = S^ — {0} in this case. In each case, Sg is separable, metrizable and locally compact, and the complements in Sg u {0} of the com- pact subsets of S0 are the open neighborhoods of 0 in Sg u {0}. For each x e A, the function /J—K^"1 (#'))(^(*))> which is the composition of the Gelfand transformation 9Ug(x) and co""1, is continuous on S£, and its restric- tion to S^ is x. We denote this function also by x, by abuse of notation. When Q & S'g we have co™x(0) = cjo and hence Jc(0) = 0, which shows that x e ^c(Stf) in every case (13.20.6). To prove the density assertion in (ii), note that the Gelfand transformation is an isometry of stg onto ^c(X(*aQ) (15.4.14); the functions A + x (where 1 e C) therefore form a dense subset of *c(Sp, whence the assertion follows in all cases. (15.9.2.3) Preliminaries to the construction ofmg.o that the algebra A/ng can be identified with C, and the condition (N)Banach